
Lecture 10
Kaplan–Meier estimator
(product-limit estimator) for F
Now we can introduce the desired estimator for the distribution func-
tion F of lifetimes, based on censored observations.
Since M
n
accumulates the “purely noisy” component of N
n
, we
obtain our estimator by equating M
n
to zero, that is, as a solution to the
equation
N
n
(t) −
Z
t
0
Y
n
(y)
dF(y)
1 −F(y)
= 0, 0 ≤t ≤
e
T
(n)
, (10.1)
with respect to F. Here
e
T
(n)
is the moment of the last jump of the
process Y
n
.
For piece-wise constant function N
n
(t) with jumps, and all trajec-
tories of N
n
(t) are such functions, the solution is given by
1 −F
n
(t) =
∏
y<t
1 −
dN
n
(y)
Y
n
(y)
, 0 ≤t ≤
e
T
(n)
.
The function F
n
is called the ...